Saturation in prismatic color model

An hue C is on some line between a neutral hue N and the most saturated (pure) one P on the triangle boundary. The saturation is the fraction of the distance along the line (0 for neutral, 1 on the boundary, about 0.25 for the example below):

The lines of constant saturation are just inscribed triangles:

These are also just isolines of the smallest barycentric coordinate, which will always vary from 1/3 at N to 0 on the boundary. This suggests the saturation s is:

s = 1 - 3*min(rho, beta, gamma)

The pure color P is just at s = 1, and the color C can be expressed:

C = N + s(P-N)

Thus

P = N + (C - N)/s

Now that we know P, we can change s to a new value s' and apply the formula